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ALGORITHMS OF INFORMATICS
Volume 3
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ALGORITHMS OF INFORMATICS
Volume 3
iii
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Table of Contents
................................
................................
................................
................................
..........................
ix
................................
................................
................................
................................
............................
x
Introduction
................................
................................
................................
................................
......
xii
24. The
Branch and Bound Method
................................
................................
................................
....
1
1. 24.1 An example: the Knapsack Problem
................................
................................
.............
1
1.1. 24.1.1
The Knapsack Problem
................................
................................
..................
1
1.2. 24.1.2 A numerical example
................................
................................
.....................
3
1.3. 24.1.3 Properties in the calculation
of the numerical example
................................
.
6
1.4. 24.1.4 How to accelerate the method
................................
................................
........
7
2. 24.2 The general f
rame of the B&B method
................................
................................
.........
8
2.1. 24.2.1 Relaxation
................................
................................
................................
......
8
2.1.1. Relaxations of a particular problem
................................
.............................
8
2.1.2. Relaxing the non

algebraic constraints
................................
........................
8
2.1.3. Relaxing the algebraic constraints
................................
................................
8
2.1.4. Relaxing the objective function
................................
................................
....
9
2.1.5. The Lagrange Relaxation
................................
................................
...........
11
2.1.6. What is common in all relaxation?
................................
.............................
12
2.1.7. Relaxation of a problem class
................................
................................
....
13
2.2. 24.2.2 The general frame of the B&B method
................................
........................
14
3. 24.3 Mixed integer programming with bounded variables
................................
..................
17
3.1. 24.3.1 The geometric analysis of a numerical example
................................
..........
17
3.2. 24.3.2 The linear programming background of the
method
................................
....
20
3.3. 24.3.3 Fast bounds on lower and upper branches
................................
...................
26
3.4. 24.3.4 Branching strate
gies
................................
................................
.....................
29
3.4.1. The LIFO Rule
................................
................................
...........................
29
3.4.2. The maximal bound
................................
................................
....................
30
3.4.3. Fast bounds and estimates
................................
................................
..........
30
3.4.4. A Rule based on depth, bound, and estimates
................................
............
31
3.5. 24.3.5 The selection of the branching variable
................................
.......................
31
3.5.1. Selection based on the fractional part
................................
.........................
31
3.5.2. Selection based on fast bounds
................................
................................
...
31
3.5.3. Priority rule
................................
................................
................................
32
3.6. 24.3.6 The numerical example is revisited
................................
.............................
32
4. 24.4 On the enumeration tree
................................
................................
..............................
35
5. 24.5 The use of information obtained from other sources
................................
...................
37
5.1. 24.5.1 Application of heuristic methods
................................
................................
.
37
5.2. 24.5.2 Preprocessing
................................
................................
...............................
37
6. 24.6 Branch and Cut
................................
................................
................................
............
38
7. 24.7 Bran
ch and Price
................................
................................
................................
.........
41
25. Comparison Based Ranking
................................
................................
................................
.......
43
1. 25.1 Introduction to supertournaments
................................
................................
................
43
2. 25.2 Introduction to

tournaments
................................
................................
...............
44
3. 25.3 Existence of

tourn
aments with prescribed score sequence
................................
45
4. 25.4 Existence of an

tournament with prescribed score sequence
.............................
47
5. 25.5 Existence of an

tournament with prescribed score sequence
.............................
48
5.1. 25.5.1 Existence of a tournament with arb
itrary degree sequence
..........................
48
5.2. 25.5.2 Description of a naive reconstructing algorithm
................................
..........
49
5.3. 25.5.3 Computation of
................................
................................
.........................
49
5.4. 25.5.4 Description of a construction algorithm
................................
......................
49
5.5. 25.5.5 Computation of
and
................................
................................
...............
50
5.6. 25.5.6 Description of a t
esting algorithm
................................
...............................
51
5.7. 25.5.7 Description of an algorithm computing
and
................................
..........
52
5.8. 25.5.8 Computing of
and
in linear time
................................
............................
53
5.9. 25.5.9 Tournament with
and
................................
................................
.............
53
5.10. 25.5.10 Description of the score slicing algorithm
................................
...............
54
5.11.
25.5.11 Analysis of the minimax reconstruction algorithm
................................
..
56
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6. 25.6 Imbalances in

tournaments
................................
................................
.................
57
6.1. 25.6.1 Imbalances in

tournaments
................................
................................
.
57
6.2. 25.6.2 Imbalances in

tournaments
................................
................................
.
57
7. 25.7 Supertournaments
................................
................................
................................
........
61
7.1. 25.7.1 Hypertournaments
................................
................................
........................
61
7.2. 25.7.2 Supertournaments
................................
................................
........................
66
8. 25.8 Football
tournaments
................................
................................
................................
...
67
8.1. 25.8.1 Testing algorithms
................................
................................
.......................
67
8.1.1. L
inear time testing algorithms
................................
................................
....
67
8.2. 25.8.2 Polynomial testing algorithms of the draw sequences
................................
.
74
9. 25.9 Reconstruction of the tested sequences
................................
................................
.......
79
26. Complexity of Words
................................
................................
................................
.................
81
1. 26.1 Simple complexity measures
................................
................................
.......................
81
1.1. 26.1.1 Finite words
................................
................................
................................
.
81
1.2. 26.1.2 Infinite words
................................
................................
...............................
82
1.3. 26.1.3 Word graphs
................................
................................
................................
84
1.3.1. De Bruijn graphs
................................
................................
........................
84
1.3.2. Algorithm to generate De Bruijn words
................................
.....................
86
1.3.3. Rauzy graphs
................................
................................
..............................
87
1.4. 26.1.4 Complexity of words
................................
................................
...................
89
1.4.1. Subword complexity
................................
................................
..................
89
1.4.2. Maximal complexity
................................
................................
..................
91
1.4.3. Global maximal complexity
................................
................................
.......
92
1.4.4. Total complexity
................................
................................
........................
95
2. 26.2 Generalized complexity measures
................................
................................
...............
97
2.1. 26.2.1 Rainbow words
................................
................................
............................
97
2.1.1. The case
................................
................................
........................
100
2.1.2. The case
................................
................................
.................
102
2.2. 26.2.2 General words
................................
................................
............................
105
3. 26.3 Palindrome complexity
................................
................................
.............................
105
3.1. 26.3.1 Palindromes in finite words
................................
................................
.......
105
3.2
. 26.3.2 Palindromes in infinite words
................................
................................
....
107
3.2.1. Sturmian words
................................
................................
........................
108
3.2.2. Power word
................................
................................
..............................
108
3.2.3. Champernowne word
................................
................................
...............
108
27. Conflict Situations
................................
................................
................................
....................
112
1. 27.1 The basics of multi

objective programming
................................
.............................
112
1.1. 27.1.1 Applications of utility functions
................................
...............................
115
1.2. 27.1.2 Weighting method
................................
................................
.....................
117
1.3. 27.1.3 Distance

dependent methods
................................
................................
.....
118
1.4. 27.1.4 Direction

dependent methods
................................
................................
....
120
2. 27.2 Method of equilibrium
................................
................................
.............................
123
3. 27.3 Methods of cooperative games
................................
................................
..................
126
4. 27.4 Collective decision

making
................................
................................
.......................
129
5. 27.5 Applications of Pareto games
................................
................................
...................
134
6. 27.6 Axiomatic methods
................................
................................
................................
...
136
28. General Purpose Computing on Graphics Processing Units
................................
.....................
142
1. 28.1 The graphics pipeline model
................................
................................
.....................
147
1.1. 28.1.1 GPU as the implementation of incremental image synthesis
.....................
145
1.1.1. Tessellation
................................
................................
..............................
145
1.1.2. Vertex processing
................................
................................
.....................
145
1.1.3. The
geometry shader
................................
................................
................
145
1.1.4. Clipping
................................
................................
................................
....
145
1.1.5. Rasterization with linear interpolation
................................
.....................
146
1.1.6. Fragment shading
................................
................................
.....................
146
1.1.7. Merging
................................
................................
................................
....
1
46
2. 28.2 GPGPU with the graphics pipeline model
................................
................................
147
2.1. 28.2.1 Output
................................
................................
................................
........
147
2.2. 28.2.2 Input
................................
................................
................................
...........
148
2.3. 28.2.3 Functions and paramete
rs
................................
................................
..........
148
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3. 28.3 GPU as a vector processor
................................
................................
........................
149
3.1. 28.3.1 Implementing the SAXPY BLAS function
................................
................
151
3.2. 28.3.2 Image filtering
................................
................................
...........................
151
4. 28.4 Beyond vector processing
................................
................................
.........................
152
4.1. 28.4.1 SIMD or M
IMD
................................
................................
.........................
152
4.2. 28.4.2 Reduction
................................
................................
................................
...
153
4.3. 28.4.3 Implementing scatter
................................
................................
.................
154
4.4. 28.4.4 Parallelism versus reuse
................................
................................
.............
156
5. 28.5 GPGPU programming model: CUDA and OpenCL
................................
.................
157
6. 28.6 Matrix

vector multiplication
................................
................................
.....................
157
6.1. 28.6.1 Making matrix

vector multiplication more parallel
................................
...
158
7. 28.7 Case study: computational fluid dynamics
................................
................................
160
7.1. 28.7.1 Eulerian solver for fluid dynamics
................................
.............................
162
7.1.1. Advection
................................
................................
................................
.
162
7.1.2. Diffusion
................................
................................
................................
..
162
7.1.3. External force field
................................
................................
...................
163
7.1.4. Projection
................................
................................
................................
.
163
7.1.5. Eulerian simulation on the GPU
................................
...............................
164
7.2. 28
.7.2 Lagrangian solver for differential equations
................................
..............
166
7.2.1. Lagrangian solver on the GPU
................................
................................
.
168
29. Perfect Arrays
................................
................................
................................
...........................
171
1. 29.1 Basic concepts
................................
................................
................................
...........
171
2. 29.2 Necessary condition and earlier results
................................
................................
.....
173
3. 29.3 One

dimensional perfect arrays
................................
................................
.................
173
3.1. 29.3.1 Pseudocode of the algorithm
Quick

Martin
................................
............
173
3.2. 29.3.2 Pseudocode of the algorithm
Optimal

Martin
................................
........
174
3.3. 29.3.3 Pseudocode of the algorithm
Shift
................................
..........................
174
3.4. 29.3.4 Pseudocode of the algorithm
Even
................................
............................
174
4. 29.4 Two

dimensional perfect arrays
................................
................................
................
174
4.1. 29.4.1 Pseudocode of the algorithm
Mesh
................................
............................
175
4.2. 29.4.2 Pseudocode of the algorithm
Cellular
................................
....................
175
5. 29.5 Three

dimensional perfect arrays
................................
................................
..............
176
5.1. 29.5.1 Pseudocode of the algorithm
Colour
................................
........................
176
5.2. 29.5.2 Pseudocode of the algorithm
Growin
g
................................
......................
176
6. 29.6 Construction of growing arrays using colouring
................................
.......................
177
6.1. 29.6.1 Construction of
growing sequences
................................
...........................
177
6.2. 29.6.2 Construction of growing squares
................................
...............................
177
6.3. 29.6.3 Construction
of growing cubes
................................
................................
..
179
6.4. 29.6.4 Construction of a four

dimensional double hypercube
..............................
179
7. 2
9.7 The existence theorem of perfect arrays
................................
................................
....
180
8. 29.8 Superperfect arrays
................................
................................
................................
....
182
9. 29.9

complexity of one

dimensional arrays
................................
................................
...
182
9.1. 29.9.1 Definitions
................................
................................
................................
.
182
9.2. 29.9.2
Bounds of complexity measures
................................
................................
184
9.3. 29.9.3 Recurrence relations
................................
................................
..................
186
9
.4. 29.9.4 Pseudocode of the algorithm
Quick

Martin
................................
............
187
9.5. 29.9.5 Pseudocode of algorithm

Complexity
................................
...................
188
9.6. 29.9.6 Pseudocode of algorithm
Super
................................
................................
188
9.7. 29.9.7 Pseudocode of algorithm
MaxSub
................................
..............................
188
9.8. 29.9.8 Construction and complexity of extremal words
................................
.......
189
10. 29.10 Finite two

dimensional arrays with maximal complexity
................................
.....
192
10.1. 29.10.1 Definitions
................................
................................
.............................
192
10.2. 29.10.2 Bounds of complexity functions
................................
............................
193
10.3. 29.10.3 Properties of the maximal complexity function
................................
.....
194
10.4. 29.10.4 On the existence of maximal arrays
................................
.......................
195
30. Score Sets and Kings
................................
................................
................................
................
198
1. 30.1 Score sets in 1

tournaments
................................
................................
.......................
199
1.1. 30.1.1 Determining the score set
................................
................................
..........
199
1.2. 30.1.2 Tournaments with prescribed score set
................................
......................
200
1.2.1. Correctness of the algorithm
................................
................................
....
203
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1.2.2. Computational complexity
................................
................................
.......
205
2. 30.2 Score sets in oriented graphs
................................
................................
.....................
205
2.1. 30.2.1 Oriented graphs with prescribed scoresets
................................
.................
207
2.1.1. Algorithm description
................................
................................
..............
208
2.1.2. Algorithm description
................................
................................
..............
209
3. 30.3 Unic
ity of score sets
................................
................................
................................
..
210
3.1. 30.3.1 1

unique score sets
................................
................................
.....................
211
3.2. 30.3.2 2

unique score sets
................................
................................
.....................
212
4. 30.4 Kings and serfs in tournaments
................................
................................
.................
214
5. 30.5 Weak kings in oriented graphs
................................
................................
..................
220
Bibliography
................................
................................
................................
................................
...
229
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List of Figures
24.1. The first seven steps of the solution.
................................
................................
..........................
3
24.2. The geometry of linear programming relaxation of Problem (24.36) including the feasible region
(triangle
), the optimal solution (
), and the optimal level of the objective function
represented by the line
.
................................
................................
............................
18
24.3. The geometry of the course of the solution. The co

ordinates of the points are: O=(0,0), A=(0,3),
B=(2.5,1.5), C=(2,1.8), D=(2,1.2), E=(
,2), F=(0,2), G=(
,1), H=(0,1
), I=(1,2.4), and J=(1,2). The feasible
regions of the relaxation are as follows. Branch 1:
, Branch 2:
, Branch 3: empty set, Branch
4:
, Branch 5:
, Branch 6:
, Branch 7: empty set (not on the figure). Point J is the
optimal solution.
................................
................................
................................
...............................
18
24.4. The course of the solution of Problem (24.36). The upper numbers in the circuits are explained in
subsection 24.3.2. They are the corrections of the previous bounds obtained fr
om the first pivoting step of
the simplex method. The lower numbers are the (continuous) upper bounds obtained in the branch.
19
24.5. The elements of the dual simplex tab
leau.
................................
................................
...............
23
25.1. Point matrix of a chess+last trick

bridge tournament with
players.
..............................
44
25.2. Point matrix of a

tournament with
for
.
...................
51
25.3. The point table of a

tournament
.
................................
................................
...........
55
25.4. The point table of
reconstructed by
Score

Slicing
.
................................
.........................
55
2
5.5. The point table of
reconstructed by
Mini

Max
.
................................
................................
...
56
25.6. Number of binomial, head halfing and good sequences, further the ratio of the numbers of good
s
equences for neighbouring values of
.
................................
................................
..........................
77
25.7. Sport table belonging to the sequence
.
................................
............................
78
26.1. The De Bruijn graph
.
................................
................................
................................
..
84
26.2. The De Bruijn graph
.
................................
................................
................................
..
85
26.3. The De Bruijn tree
.
................................
................................
................................
..
87
26.4. Rauzy graphs for the infinite Fibonacci word.
................................
................................
.........
88
26.5. Rauzy graphs for the power word.
................................
................................
...........................
88
26.6. Complexity of several binary words.
................................
................................
.......................
91
26.7. Complexity of all 3

length binary words.
................................
................................
................
92
26.8. Complexity of all 4

length bina
ry words.
................................
................................
................
93
26.9. Values of
,
, and
.
................................
................................
............................
93
26.10. Frequency
of words with given total complexity.
................................
................................
..
99
26.11. Graph for

subwords when
.
................................
................................
.................
98
26.12.

complexity for rainbow words of length 6 and 7.
................................
.....................
99
26.13. The

complexity of words of length
.
................................
................................
........
101
26.14. Values of
.
................................
................................
................................
...............
103
27.1. Planning of a sewage plant.
................................
................................
................................
...
113
27.2. The image of set
.
................................
................................
................................
...............
114
27.3. The image of set
.
................................
................................
................................
...............
114
27.4. Minimizing distance.
................................
................................
................................
.............
118
27.5. Maximizing distance.
................................
................................
................................
.............
119
27.6. The image of the normalized set
.
................................
................................
......................
119
27.7. Direction

dependent methods.
................................
................................
...............................
121
27.8. The graphical solution of Example 27.6.
................................
................................
...............
121
27.9. The graphical solution of Example 27.7.
................................
................................
...............
122
27.10. Game with no equilibrium.
................................
................................
................................
..
124
27.11. Group decision

making table.
................................
................................
..............................
131
27.12. The database of Example 27.17.
................................
................................
..........................
132
27.13. The preference graph of Example 27.17.
................................
................................
.............
133
27.14. Group decision

making table.
................................
................................
..............................
134
27.15. Group decision

making table.
................................
................................
..............................
134
27.16. The database of Example 27.18.
................................
................................
..........................
135
27.17.
................................
................................
................................
................................
.............
136
27.18. Kalai
–
Smorodinsky solution.
................................
................................
...............................
138
27.19. Solution of Example 27.20.
................................
................................
................................
.
138
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27.20. The method of monotonous area.
................................
................................
........................
139
28.1. GPU programming models for shader APIs and for CUDA. We depict here a Shader Model 4
c
ompatible GPU. The programmable stages of the shader API model are red, the fixed

function stages are
green.
................................
................................
................................
................................
..............
142
28.2. Incremental image synthesis process.
................................
................................
....................
144
28.3. Blending unit that computes the new pixel color of the frame buffer as a function of its old color
(destination) and the new fragment color (source).
................................
................................
........
147
28.4. GPU as a vector processor.
................................
................................
................................
....
149
28.5. An example for paralle
l reduction that sums the elements of the input vector.
.....................
153
28.6. Implementation of scatter.
................................
................................
................................
.....
154
28.7. Caustics rendering is a practical use of histogram generation. The illumination intensity of the target
will be proportional to the number of photons it receives (images courtesy of Dávid Balambér).
.
156
28.8. Finite element representations of functions. The texture filtering of the GPU directly supports finite
element representations using regularly placed samples in one

, two

, and three

dimensions
and
interpolating with piece

wise constant and piece

wise linear basis functions.
...............................
160
28.9. A time step of the Eulerian solver updates textures encoding the velocity
field.
..................
164
28.10. Computation of the simulation steps by updating three

dimensional textures. Advection utilizes the
texture filtering hardware. The linear equations of t
he viscosity damping and projection are solved by
Jacobi iteration, where a texel (i.e. voxel) is updated with the weighted sum of its neighbors, making a
single Jacobi iteration step equivalent to an image filtering operation.
................................
..........
164
28.11. Flattened 3D velocity (left) and display variable (right) textures of a simulation.
..............
165
28.12. Snapshots from an animation rendered with Eulerian fluid dynamics.
................................
165
28.13. Data structures stored in arrays or textures. One

dimensional float3 arrays
store the particles'
position and velocity. A one

dimensional float2 texture stores the computed density and pressure. Finally,
a two

dimensional texture identifies nearby particles for each particle.
................................
.........
168
28.14. A time step of the Lagrangian solver. The considered particle is the red one, and its neighbors are
yellow.
................................
................................
................................
................................
............
168
28.15. Animations obt
ained with a Lagrangian solver rendering particles with spheres (upper image) and
generating the isosurface (lower image) [
114
].
................................
................................
..............
168
29.
1. a) A (2,2,4,4)

square;
b) Indexing scheme
of size
................................
..........
178
29.2. Binary colouring matrix
of size
................................
................................
................
178
29.3. A (4,2,2,16)

square generated by colouring
................................
................................
..........
178
29.4. A (2,2,4,4)

square
................................
................................
................................
..................
180
29.5. Sixteen layers of the

perfect output of
Shift
................................
.....................
180
29.6. Values of jumping function of rainbow words
of length
.
................................
...
191
30.1. A round

robin competition involving 4 players.
................................
................................
....
198
30.2. A tournament with score set
.
................................
................................
.......................
199
30.3. Out

degree matrix of the tournament represented in Figure 30.2.
................................
.........
199
30.4. Construction of tournament
with odd number of distinct scores.
................................
......
201
30.5. Construction of tournament
with even number of distinct scores.
................................
.....
201
30.6. Out

degree matrix of the tournament
.
................................
................................
............
202
30.7. Out

degree matrix of the tournament
.
................................
................................
............
202
30.8. Out

degree
matrix of the tournament
.
................................
................................
...............
203
30.9. An oriented graph with score sequence
and score set
.
.............................
205
30.10. A tournament with three kings
and three serfs
. Note that
is neither a king nor a
serf and
are both kings and serfs.
................................
................................
..........................
215
30.11. A tournament with three kings and two strong kings
................................
..........................
217
30.12. Construction of an

tourn
ament with even
.
................................
..............................
218
30.13. Six vertices and six weak kings.
................................
................................
..........................
220
30.14. Six vertic
es and five weak kings.
................................
................................
.........................
221
30.15. Six vertices and four weak kings.
................................
................................
........................
221
30.16.
Six vertices and three weak kings.
................................
................................
.......................
221
30.17. Six vertices and two weak kings.
................................
................................
.........................
222
30.18. Vertex of maximum score is not a king.
................................
................................
..............
222
30.19. Construction of an

oriented graph.
................................
................................
.....
225
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AnTonCom,
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This electronic
book was
prepared in the
framework of
project Eastern
Hungarian
Informatics
Books
Repository no.
TÁMOP

4.1.2

08/1/A

2009

0046
This electronic
book appeared
with the support
of European
Union and with
the co

financing
of European
Social Fund
Nemzeti
Fejlesztési
Üg
ynökség
http://ujszecheny
iterv.gov.hu/
06
40 638

638
Editor:
Antal
Iványi
Authors of
Volume 3:
Béla
Vizvári
(Chapter
24), Antal
Iványi
and Shariefuddin
Pirzada
(Chapter
25), Mira

Cristiana
Anisiu
and Zoltán
Kása
(Chapter 26),
Ferenc
Szidarovszky
and
László
Domoszlai
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(Chapter 27),
László
Szirmay

Kalos
and László
Szécsi
(Chapter
28), Antal
Iványi
(Chapter 29),
Shariefuddin
Pirzada
, Antal
Iványi
and
Muhammad Ali
Khan
(Chapter
30)
Validators of
Volume 3:
György Kovács
(Chapter 24),
Zoltán
Kása
(Chapter 25),
Antal
Iványi
(Chapter 26),
Sándor
Molnár
(Chapter 27),
György
Antal
(Chapter 28),
Zoltán
Kása
(Chapter 29),
Zoltán
Kása
(Chapter 30),
Anna
Iványi
(Bibliography)
©2011
AnTonCom
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Introduction
The third volume contains seven new chapters.
Chapter 24 (
The Branch and Bound Method
) was written by Béla Vizvári (Eastern Mediterrean University),
Chapter 25 (
Comparison Based Ranking
) by Antal Iványi (Eötvös Loránd University) and Shariefuddin Pirzada
(University of Kashmir), Chapter 26 (
Complexity of Words
) b
y Zoltán Kása (Sapientia Hungarian University of
Transylvania) and Mira

Cristiana Anisiu (Tiberiu Popovici Institute of Numerical Mathematics), Chapter 27
(
Conflict Situations
) by Ferenc Szidarovszky (University of Arizona) and by László Domoszlai (Eötvös
Loránd
University), Chapter 28 (
General Purpose Computing on Graphics Processing Units
) by László Szirmay

Kalos
and László Szécsi (both Budapest University of Technology and Economics), Chapter 29 (
Perfect Arrays
) by
Antal Iványi (Eötvös Loránd University)
, and Chapter 30 by Shariefuddin Pirzada (University of Kashmir),
Antal Iványi (Eötvös Loránd University) and Muhammad Ali Khan (King Fahd University).
The LaTeX style file was written by Viktor Belényesi, Zoltán Csörnyei, László Domoszlai and Antal
Iványi.
The figures was drawn or corrected by Kornél Locher and László Domoszlai. Anna Iványi transformed the
bibliography into hypertext. The DOCBOOK version was made by Marton 2001. Kft.
Using the data of the colofon page you can contact with any of the
creators of the book. We welcome ideas for
new exercises and problems, and also critical remarks or bug reports.
The publication of the printed book (Volumes 1 and 2) was supported by Department of Mathematics of
Hungarian Academy of Science. This electron
ic book (Volumes 1, 2, and 3) was prepared in the framework of
project
Eastern Hungarian Informatics Books Repository
no. TÁMOP

4.1.2

08/1/A

2009

0046. This electronic
book appeared with the support of European Union and with the co

financing of European S
ocial Fund.
Budapest, September 2011
Antal Iványi (
tony@compalg.inf.elte.hu
)
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Chapter
24.
The Branch and Bound
Method
It
has serious practical consequences if it is known that a combinatorial problem is NP

complete. Then one can
conclude according to the present state of science that no simple combinatorial algorithm can be applied and
only an enumerative

type method can so
lve the problem in question. Enumerative methods are investigating
many cases only in a non

explicit, i.e. implicit, way. It means that huge majority of the cases are dropped based
on consequences obtained from the analysis of the particular numerical prob
lem. The three most important
enumerative methods are (i) implicit enumeration, (ii) dynamic programming, and (iii) branch and bound
method. This chapter is devoted to the latter one. Implicit enumeration and dynamic programming can be
applied within the f
amily of optimization problems mainly if all variables have discrete nature. Branch and
bound method can easily handle problems having both discrete and continuous variables. Further on the
techniques of implicit enumeration can be incorporated easily in t
he branch and bound frame. Branch and bound
method can be applied even in some cases of nonlinear programming.
The
Branch and Bound
(abbreviated further on as B&B) method is just a frame of a large family of methods. Its
substeps can be carried out in diff
erent ways depending on the particular problem, the available software tools
and the skill of the designer of the algorithm.
Boldface letters denote vectors and matrices; calligraphic letters are used for sets. Components of vectors are
denoted by the same
but non

boldface letter. Capital letters are used for matrices and the same but lower case
letters denote their elements. The columns of a matrix are denoted by the same boldface but lower case letters.
Some formulae with their numbers are repeated severa
l times in this chapter. The reason is that always a
complete description of optimization problems is provided. Thus the fact that the number of a formula is
repeated means that the formula is
identical
to the previous one.
1.
24.1 An example: the Knapsack
Problem
In this section the branch and bound method is shown on a numerical example. The problem is a sample of the
binary knapsack problem which is one of the easiest problems of integer programming but it is still NP

complete. The calculations are carri
ed out in a brute force way to illustrate all features of B&B. More intelligent
calculations, i.e. using implicit enumeration techniques will be discussed only at the end of the section.
1.1.
24.1.1 The Knapsack Problem
There are many different knapsack pr
oblems. The first and classical one is the binary knapsack problem. It has
the following story. A tourist is planning a tour in the mountains. He has a lot of objects which may be useful
during the tour. For example ice pick and can opener can be among the
objects. We suppose that the following
conditions are satisfied.
•
Each object has a positive value and a positive weight. (E.g. a balloon filled with helium has a negative
weight. See Exercises
24.1

1
[
7
]
and
24.1

2
[
8
]
) The value is the degree of contribution of the object to the
success of the tour.
•
The objects are
independent from each other. (E.g. can and can opener are not independent as any of them
without the other one has limited value.)
•
The knapsack of the tourist is strong and large enough to contain all possible objects.
•
The strength of the tourist makes pos
sible to bring only a limited total weight.
•
But within this weight limit the tourist want to achieve the maximal total value.
The following notations are used to the mathematical formulation of the problem:
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For each object
a so

called
binary
or
zero

on
e
decision variable, say
, is introduced:
Notice that
is the weight of the object in the knapsack.
Similarly
is the value of the object on the tour. The total weight in the knapsack is
which may not exceed the weight limit. Hence the mathematical
form of the problem is
The difficulty of the problem is caused by the integrality requirement. If constraint (
24.3
) is substituted by the
relaxed constraint, i.e. by
then the Problem (
24.1
), (
24.2
), and (
24.4
) is a linear programming problem. (
24.4
) means that not only a
complete object can be in the knap
sack but any part of it. Moreover it is not necessary to apply the simplex
method or any other LP algorithm to solve it as its optimal solution is described by
Theorem 24.1
Suppose that the numbers
are all positive and moreover the index order
satisfies the inequality
Then there is an index
and an optimal solution
such that
Notice that there is only at most one non

integer component in
. This property will be used at the numerical
calcula
tions.
From the point of view of B&B the relation of the Problems (
24.1
), (
24.2
), and (
24.3
) and (
24.1
), (
24.2
), and
(
24.4
) is very important. Any feasible solution of the first one is also feasible in the second one. But the opposite
statement is not true. In other words the set of feasible solutions of th
e first problem is a proper subset of the
feasible solutions of the second one. This fact has two important consequences:
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•
The optimal value of the Problem (
24.1
), (
24.2
), and (
24.4
) is an upper bound of the optimal value of the
Problem (
24.1
), (
24.2
), and (
24.3
).
•
If the optimal solution of the Problem (
24.1
), (
24.2
), and (
24.4
) is feasible in the Problem (
24.1
), (
24.2
), and
(
24.3
) then it is the optimal solution of the latter problem as well.
These properties are used in the course of the branch and bound method intensively.
1.2.
24.1.2 A numerical example
The basic technique of the B&B method is that it di
vides the set of feasible solutions into smaller sets and tries
to fathom them. The division is called
branching
as new branches are created in the enumeration tree. A subset
is fathomed if it can be determined exactly if it contains an optimal solution.
T
o show the logic of B&B the problem
will be solved. The course of the solution is summarized on Figure
24.1
.
Notice that condition (
24.5
) is satisfied as
The set of the feasible solution
s of (
24.6
) is denoted by
, i.e.
The continuous relaxation of (
24.6
) is
The set of the feasible solutions of (
24.7
) is denoted by
, i.e.
Thus the
difference between (
24.6
) and (
24.7
) is that the value of the variables must be either 0 or 1 in (
24.6
)
and on the other hand they can take any value from
the closed interval
in the case of (
24.7
).
Because Problem (
24.6
) is difficult, (
24.7
) is solved instead. The optimal solution according to Theorem
24.1
[
2
]
is
As the value of
is non

integer, the optimal value 67.54 is just an upper bound of the optimal value of (
24.6
)
and further analysis
is needed. The value 67.54 can be rounded down to 67 because of the integrality of the
coefficients in the objective function.
The key idea is that the sets of feasible solutions of both problems are divided into two parts according the two
possible values
of
. The variable
is chosen as its value is non

integer. The importance of the choice is
discussed below.
Figure
24.1.
The first seven steps of the solution.
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Let
and
Obviously
Hence the problem
is a relaxation of the problem
Problem (
24.8
) can be solved by Theorem
24.1
[
2
]
, too, but it must be taken into consideration that the value of
is 0. Thus its optimal solution is
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The
optimal value is 65.26 which gives the upper bound 65 for the optimal value of Problem (
24.9
). The other
subsets of the feasible solutions are immediately investigated. The optimal solution of the problem
is
giving
the value 67.28. Hence 67 is an upper bound of the problem
As the upper bound of (
24.11
) is higher than the upper bound of (
24.9
), i.e. this branch is more promising, first
it is fatho
med further on. It is cut again into two branches according to the two values of
as it is the non

integer variable in the optimal solution of (
24.10
). Let
The sets
and
are containing the feasible solution of the o
riginal problems such that
is fixed to 1 and
is fixed to 0. In the sets
and
both variables are fixed to 1. The optimal solution of the first relaxed
problem, i.e.
is
As it is integer it is also the optimal solution of the problem
The optimal objective function value is 65. The branch of the sets
and
is completely fathomed, i.e. it is
not possible to find a better solution in it.
The other new branch is when both
and
are fixed to 1. If the objective function is optimized on
then
the optimal solution is
Applying the same technique again two branches are defined by the sets
The optimal solution of the branch of
is
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The optimal value is 63.32. It is strictly less than the objective function value of the feasible solut
ion found in
the branch of
. Therefore it cannot contain an optimal solution. Thus its further exploration can be omitted
although the best feasible solution of the branch is still not known. The branch of
is infeasible as objects 1,
2, and 3 are overus
ing the knapsack. Traditionally this fact is denoted by using
as optimal objective
function value.
At this moment there is only one branch which is still unfathomed. It is the branch of
. The upper bound here
is 65 which is equal to the objective functi
on value of the found feasible solution. One can immediately
conclude that this feasible solution is optimal. If there is no need for alternative optimal solutions then the
exploration of this last branch can be abandoned and the method is finished. If alt
ernative optimal solutions are
required then the exploration must be continued. The non

integer variable in the optimal solution of the branch
is
. The subbranches referred later as the 7th and 8th branches, defined by the equations
and
,
give the uppe
r bounds 56 and 61, respectively. Thus they do not contain any optimal solution and the method is
finished.
1.3.
24.1.3 Properties in the calculation of the numerical example
The calculation is revisited to emphasize the general underlying logic of the met
hod. The same properties are
used in the next section when the general frame of B&B is discussed.
Problem (
24.6
) is a difficult one. Therefore the very similar but much easier Problem (
24.
7
) has been solved
instead of (
24.6
). A priori it was not possible to exclude the case that the optimal solution of (
24.7
) is the
optimal solution of (
24.6
) as well. Finally it turned out that the optimal solution of (
24.7
) does not satisfy all
constraints of (
24.6
) thus it is not optimal there. But the calculation was not useless, becau
se an upper bound of
the optimal value of (
24.6
) has been obtained. These properties are reflected in the definition of
relaxation
in the
next section.
As the relaxation did not solved Problem (
24.6
) therefore it was divided into Subproblems (
24.9
) and (
24.11
).
Both subproblems have their own optimal solution and the better one is the optimal solution of (
24.6
). They are
still too difficult to be solved directly, therefore relaxations were generated to both of them. These problems are
(
24.8
) and (
24.10
). The nature of (
24.8
) and (
24.10
) from mathematical point of view is the same as of (
24.7
).
Notice that the union of the sets of the feasible solutions of (
24.8
) and (
24.10
) is a proper subset of the relaxation
(
24.7
), i.e.
Moreover the two subsets have no common element, i.e.
It is true for all other cases, as well.
The reason is that the branching, i.e. the determination of the Subproblems
(
24.9
) and (
24.11
) was made in a way that the optimal solution of the relaxation, i.e. the optimal solution of
(
24.7
), was cut off.
The branching policy also has consequences on the upper bounds. Let
be the optimal value of the problem
where the objective function is unchanged and the set of feasible solutions is
. Using this n
otation the optimal
objective function values of the original and the relaxed problems are in the relation
If a subset
is divided into
and
then
Notice that in the current Problem (
24.12
) is always satisfied with
strict inequality
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(The values
and
were mentioned only.) If the upper bounds of a certain quantity are compared
then one can conclude that the smaller the better as it is closer to the value to be estimated. An equation similar
to (
24.12
) is true for the non

relaxed problems, i.e. if
then
but because of the difficulty of the solution of the problems, practically it is not possible to use (
24.13
) for
getting further information.
A subproblem is fathomed and no further investigation of it is needed if either
•
its integer (non

relaxed) optimal solution is obtained, like in the case of
, or
•
it is proven to be infeasible as in the case of
, or
•
its upper bound is not greater than the
value of the best known feasible solution (cases of
and
).
If the first or third of these conditions are satisfied then all feasible solutions of the subproblem are enumerated
in an implicit way.
The subproblems which are generated in the same iteration
, are represented by two branches on the enumeration
tree. They are siblings and have the same parent. Figure 24.1 visualize the course of the calculations using the
parent
–
child relation.
The enumeration tree is modified by constructive steps when new bra
nches are formed and also by reduction
steps when some branches can be deleted as one of the three above

mentioned criteria are satisfied. The method
stops when no subset remained which has to be still fathomed.
1.4.
24.1.4 How to accelerate the method
As it was mentioned in the introduction of the chapter, B&B and implicit enumeration can co

operate easily.
Implicit enumeration uses so

called tests and obtains consequences on the values of the variables. For example
if
is fixed to 1 then the knapsack
inequality immediately implies that
must be 0, otherwise the capacity of
the tourist is overused. It is true for the whole branch 2.
On the other hand if the objective function value must be at least 65, which is the value of the found feasible
solution
then it possible to conclude in branch 1 that the fifth object must be in the knapsack, i.e.
must be 1,
as the total value of the remaining objects 1, 2, and 4 is only 56.
Why such consequences accelerate the algorithm? In the example there are 5 binary
variables, thus the number
of possible cases is
. Both branches 1 and 2 have 16 cases. If it is possible to determine the value of a
variable, then the number of cases is halved. In the above example it means that only 8 cases remain to be
investigated in
both branches. This example is a small one. But in the case of larger problems the acceleration
process is much more significant. E.g. if in a branch there are 21 free, i.e. non

fixed, variables but it is possible
to determine the value of one of them the
n the investigation of 1
048
576 cases is saved. The application of the
tests needs some extra calculation, of course. Thus a good trade

off must be found.
The use of information provided by other tools is further discussed in Section
24.5
.
Exercises
24.1

1 What is the suggestion of the optimal solution of a Knapsack Problem in connection of an object having
(a) negative weight and positive value, (b) positive weight and negative value?
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24.1

2 Show that an object of a knapsack
problem having negative weight and negative value can be substituted
by an object having positive weight and positive value such that the two knapsack problems are equivalent.
(
Hint.
Use complementary variable.)
24.1

3 Solve Problem (
24.6
) with a branching strategy such that an integer valued variable is used for branching
provided that such a variable exists.
2.
24.2 The general frame of the B&B method
The aim of this section is to give a general description of the B&B met
hod. Particular realizations of the general
frame are discussed in later sections.
B&B is based on the notion of
relaxation
. It has not been defined yet. As there are several types of relaxations
the first subsection is devoted to this notion. The general
frame is discussed in the second subsection.
2.1.
24.2.1 Relaxation
Relaxation is discussed in two steps. There are several techniques to define relaxation to a particular problem.
There is no rule for choosing among them. It depends on the design of the a
lgorithm which type serves the
algorithm well. The different types are discussed in the first part titled “Relaxations of a particular problem”. In
the course of the solution of Problem (
24.6
) subproblems were generated wh
ich were still knapsack problems.
They had their own relaxations which were not totally independent from the relaxations of each other and the
main problem. The expected common properties and structure is analyzed in the second step under the title
“Relaxa
tion of a problem class”.
2.1.1.
Relaxations of a particular problem
The description of Problem (
24.6
) consists of three parts: (1) the objective function, (2) the algebraic constraints,
and (3) the requirement that the v
ariables must be binary. This structure is typical for optimization problems. In
a general formulation an optimization problem can be given as
2.1.2.
Relaxing the non

algebraic constraints
The underlying logic of generating relaxation (
24.7
) is that constraint (
24.16
) has been substituted by a looser
one. In the particular case it was allowed that the variables can take any value between 0 and 1. In general
(
24.16
) is replaced by a requirement that the variables must belong to a set, say
, which is larger than
, i.e.
the relation
must hold. More formally the relaxation of Problem (
24.14
)

(
24.16
) is the problem
(
24.14
)
(
24.15
)
This type of relaxation can be applied if a large amount of difficulty can be eliminated by changing the nature of
the variables.
2.1.3.
Relaxing the algebraic constraints
The
re is a similar technique such that (
24.16
) the inequalities (
24.15
) are relaxed instead of the constraints. A
natural way of this type of relaxation is the following. Assume that there
are
inequalities in (
24.15
). Let
be fixed numbers. Then any
satisfying (
24.15
) also satisfies the inequality
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Then the relaxation is the optimization of the (
24.14
) objective function under the conditions (
24.18
) and
(
24.16
). The name of the inequality (
24.18
) is
surrogate co
nstraint
.
The problem
is a general zero

one optimization problem. If
then the relaxation obtained in this way is
Problem (
24.6
). Both problems belong to NP

complete classes. However the knapsack problem is significantl
y
easier from practical point of view than the general problem, thus the relaxation may have sense. Notice that in
this particular problem the optimal solution of the knapsack problem, i.e. (1,0,1,1,0), satisfies the constraints of
(
24.19
), thus it is also the optimal solution of the latter problem.
Surrogate constraint is not the only option in relaxing the algebraic constraints. A region defined by nonlinear
boundary surfaces can be approximated by tangent planes. For
example if the feasible region is the unit circuit
which is described by the inequality
can be approximated by the square
If the optimal solution on the enlarged region is e.g. the point (1,1) which is not in the original feasible region
then a
cut
mus
t be found which cuts it from the relaxed region but it does not cut any part of the original feasible
region. It is done e.g. by the inequality
A new relaxed problem is defined by the introduction of the cut. The method is similar to one of the method
r
elaxing of the objective function discussed below.
2.1.4.
Relaxing the objective function
In other cases the difficulty of the problem is caused by the objective function. If it is possible to use an easier
objective function, say
, but to obtain an uppe
r bound the condition
must hold. Then the relaxation is
(
24.15
)
(
24.16
)
This type of relaxation is typical if B&B is applied in (continuous) nonlinear optimization. An important
subclass of the nonlinear optimization problems
is the so

called convex programming problem. It is again a
relatively easy subclass. Therefore it is reasonable to generate a relaxation of this type if it is possible. A
Problem (
24.14
)

(
24.16
) is a convex programming problem, if
is a convex set, the functions
are convex and the objective function
is concave. Thus the relaxation can be a convex
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programming problem if only the last condition is violated. Then it is enough to find
a concave function
such that (
24.20
) is satisfied.
For example the single variable function
is not concave in the interval
.
Footnote.
A continuous function is concave if its second derivative is negative.
which i
s
positive in the open interval
.
Thus if it is the objective function in an optimization problem it might be necessary that it is substituted by a
concave function
such that
. It is easy to see that
satisfies
the requirements.
Let
be the optimal s
olution of the relaxed problem (
24.21
), (
24.15
), and (
24.16
). It solves the original
problem if the optimal solution has the same objective function va
lue in the original and relaxed problems, i.e.
.
Another reason why this type of relaxation is applied that in certain cases the objective function is not known in
a closed form, however it can be determined in any given point. It might happen even in the
case if the objective
function is concave. Assume that the value of
is known in the points
. If
concave then it is
smooth, i.e. its gradient exists. The gradient determines a tangent plane which is above the function. The
equation of the tangent plane in point
is
Footnote.
The gradient is considered being a row vector.
Hence in all points of the
domain of the function
we have that
Obviously the function
is an approximation of function
.
The idea if the method is illustrated on the following numerical example. Assume that an “unknown” concave
function is to be maximized on the [0,5] closed
interval. The method can start from any point of the interval
which is in the feasible region. Let 0 be the starting point. According to the assumptions although the closed
formula of the function is not known, it is possible to determine the values of fun
ction and its derivative. Now
the values
and
are obtained. The general formula of the tangent line in the point
is
Hence the equation of the first tangent line is
giving the first optimization problem as
As
is a monotone increasing function, the optimal solution is
. Then the values
and
are provided by the method calculating the function. The equation of the second tangent line is
. Thus the second optimization problem is
As the second tangent line i
s a monotone decreasing function, the optimal solution is in the intersection point of
the two tangent lines giving
. Then the values
and
are calculated and the
equation of the tangent line is
. The next optimization problem is
The Branch and Bound Method
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.
The optimal solution
is
. It is the intersection point of the first and third tangent lines. Now both new
intersection points are in the interval [0,5]. In general some intersection points can be infeasible. The method
goes in the same way further on. The approximated “unkno
w” function is
.
2.1.5.
The Lagrange Relaxation
Another relaxation called
Lagrange relaxation
. In that method both the objective function and the constraints
are modified. The underlying idea is the following. The variables must satisfy two different
types of constraints,
i.e. they must satisfy both (
24.15
) and (
24.16
). The reason that the constraints are written in two parts is that the
nature of the two sets of constraints is diffe
rent. The difficulty of the problem caused by the requirement of
both
constraints. It is significantly easier to satisfy only one type of constraints.
So what about to eliminate one of
them?
Assume again that the number of inequalities in (
24.15
) is
. Let
be fixed numbers. The
Lagrange relaxation of the problem (
24.14
)
–
(
24.16
) is
(
24.16
)
Notice that the objective function (
24.22
) penalizes the violation of the constraints, e.g. trying to use too much
resources, and rewards the sav
ing of resources. The first set of constraints disappeared from the problem. In
most of the cases the Lagrange relaxation is a much easier one than the original problem. In what follows
Problem (
24.14
)
–
(
24.16
) is also denoted by
and the Lagrange relaxation is referred as
. The notation
reflects the fact that the Lagrange relaxation problem depends on the choice of
's. The numbers
's are called
Lagrange multipliers.
It is not obvious
that
is really a relaxation of
. This relation is established by
Theorem 24.2
Assume that both
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